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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eliminate from (general) A to (upper triangular) U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Demonstrate Gaussian Elimination (page 51 of GS)\n",
    "\n",
    "(Reminder: Julia programming syntax will not be tested in this class, but that doesn't mean you can't learn from\n",
    "watching the code being executed)\n",
    "\n",
    "(If you are familiar with Jupyter notebooks and wish to execute yourself, you can download this notebook \n",
    "using the download button in the upper right, but WARNING: right click to \"download as\", or OPTION-click (MAC) or ALT-Click (Linux and Windows I think) on the download icon, or you won't get an ipynb file.) You can then drag into Jupyter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "lookat (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# nice viz for matrices\n",
    "function lookat(A; redrow=0, rounding=2, showtext=true)\n",
    "  n = size(A,1)\n",
    "  plot(legend=false, axis=false)\n",
    "  rowcolor = redrow > 0 ? :red : :black  \n",
    "  for i=1:n, j=1:n  \n",
    "      scatter!( [j],[i], ann=  showtext ? (j,i,round(A[i,j],digits=rounding), :white ) : (j,i,\"\") ,\n",
    "                color=abs(A[i,j]) > .0001 ? (i==redrow ? rowcolor : :black) : :white, \n",
    "                marker=:square, markersize=30, aspectratio=1, yflip=true, yaxis=[.5,n+.5],xaxis=[.5,n+.5])\n",
    "  end\n",
    "  plot!()\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 4.0  9.0  4.0  9.0\n",
       " 4.0  7.0  6.0  6.0\n",
       " 2.0  9.0  8.0  6.0\n",
       " 7.0  1.0  6.0  8.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(1.0:9,4,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0\n",
       " 0.0  0.0  0.0  0.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = fill(0.0,4,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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   "source": [
    "lookat(A)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What multiple of row 1 must we subtract from row 2 to zero out the (2,1) entry?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L[2,1] = A[2,1] / A[1,1]"
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  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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    "# A[2,:] = A[2,:] - L[2,1] * A[1,:]\n",
    "A[2,:] -= L[2,1] * A[1,:] # subtract that multiple of the first row from the second\n",
    "lookat(A, redrow=2)"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What multiple of row 1 must we subtract from row 3 to zero out the (3,1) entry?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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       "0.5"
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     "execution_count": 8,
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   ],
   "source": [
    "L[3,1] = A[3,1] / A[1,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
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   "source": [
    "A[3,:] -= L[3,1] * A[1,:]\n",
    "lookat(A, redrow=3)"
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    "L[4,1] = A[4,1] / A[1,1]\n",
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   "cell_type": "code",
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   "source": [
    "n = 5\n",
    "A = randn(n,n)\n",
    "L = fill(0.0,n,n)\n",
    "display(lookat(A))\n",
    "for j=1:n, i=(j+1):n\n",
    "        L[i,j] = A[i,j]/A[j,j]\n",
    "        A[i,:] -= L[i,j] * A[j,:]\n",
    "        display(lookat(A,redrow=i))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 5\n",
    "A = randn(n,n)\n",
    "L = fill(0.0,n,n)\n",
    "Akeep = [copy(A)]\n",
    "row = [0]\n",
    "display(lookat(A))\n",
    "for j=1:n, i=(j+1):n\n",
    "        L[i,j] = A[i,j]/A[j,j]\n",
    "        A[i,:] -= L[i,j] * A[j,:]\n",
    "        push!(Akeep,copy(A))\n",
    "        push!(row,i)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    } else if (window.location.pathname.includes(\"/lab\")) {\n",
       "        // Guessing JupyterLa\n",
       "        console.log(\"Jupyter Lab detected; make sure the @webio/jupyter-lab-provider labextension is installed.\");\n",
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       "\n",
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       "    style=\"padding: 1em; font-weight: bold;\"\n",
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       "    Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\n",
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       "    the WebIO/IJulia documentation</a>.\n",
       "    <!-- TODO: link to installation docs. -->\n",
       "</p>\n"
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       "HTML{String}(\"<script>\\n// Immediately-invoked-function-expression to avoid global variables.\\n(function() {\\n    var warning_div = document.getElementById(\\\"webio-warning-2387978101050626103\\\");\\n    var hide = function () {\\n        var script = document.getElementById(\\\"webio-setup-14795050379659235485\\\");\\n        var parent = script && script.parentElement;\\n        var grandparent = parent && parent.parentElement;\\n        if (grandparent) {\\n            grandparent.style.display = \\\"none\\\";\\n        }\\n        warning_div.style.display = \\\"none\\\";\\n    };\\n    if (typeof Jupyter !== \\\"undefined\\\") {\\n        console.log(\\\"WebIO detected Jupyter notebook environment.\\\");\\n        // Jupyter notebook.\\n        var extensions = (\\n            Jupyter\\n            && Jupyter.notebook.config.data\\n            && Jupyter.notebook.config.data.load_extensions\\n        );\\n        if (extensions && extensions[\\\"webio-jupyter-notebook\\\"]) {\\n            // Extension already loaded.\\n            console.log(\\\"Jupyter WebIO nbextension detected; not loading ad-hoc.\\\");\\n            hide();\\n            return;\\n        }\\n    } else if (window.location.pathname.includes(\\\"/lab\\\")) {\\n        // Guessing JupyterLa\\n        console.log(\\\"Jupyter Lab detected; make sure the @webio/jupyter-lab-provider labextension is installed.\\\");\\n        hide();\\n        return;\\n    }\\n})();\\n\\n</script>\\n<p\\n    id=\\\"webio-warning-2387978101050626103\\\"\\n    class=\\\"output_text output_stderr\\\"\\n    style=\\\"padding: 1em; font-weight: bold;\\\"\\n>\\n    Unable to load WebIO. Please make sure WebIO works for your Jupyter client.\\n    For troubleshooting, please see <a href=\\\"https://juliagizmos.github.io/WebIO.jl/latest/providers/ijulia/\\\">\\n    the WebIO/IJulia documentation</a>.\\n    <!-- TODO: link to installation docs. -->\\n</p>\\n\")"
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     "metadata": {},
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   ],
   "source": [
    "using Interact"
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  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
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            "id": "17084213940907812968",
            "imports": {
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               "name": "knockout_punches",
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               "url": "/assetserver/df29ad18e8884e98d1d36dd0c339f642f729bf96-knockout_punches.js"
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              {
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       "            window,\n",
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       "                '<p>Please read ' +\n",
       "                '<a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">the troubleshooting guide</a> ' +\n",
       "                'for more information on how to resolve this issue.</p>' +\n",
       "                '<p><a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/</a></p>' +\n",
       "                '</div>'\n",
       "            );\n",
       "    }\n",
       "    </script>\n",
       "</div>\n"
      ],
      "text/plain": [
       "Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Scope(Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :label), Any[nothing], Dict{Symbol,Any}(:className => \"interact \",:style => Dict{Any,Any}(:padding => \"5px 10px 0px 10px\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-left\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :input), Any[], Dict{Symbol,Any}(:max => 11,:min => 1,:attributes => Dict{Any,Any}(:type => \"range\",Symbol(\"data-bind\") => \"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\" => \"horizontal\"),:step => 1,:className => \"slider slider is-fullwidth\",:style => Dict{Any,Any}()))], Dict{Symbol,Any}(:className => \"interact-flex-row-center\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes => Dict(\"data-bind\" => \"text: formatted_val\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-right\"))], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\")), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"changes\" => (Observable{Int64} with 1 listeners. Value:\n",
       "0, nothing),\"index\" => (Observable{Int64} with 2 listeners. Value:\n",
       "1, nothing)), Set(String[]), nothing, Asset[Asset(\"js\", \"knockout\", \"/Users/alanedelman/.julia/packages/Knockout/1sDlc/src/../assets/knockout.js\"), Asset(\"js\", \"knockout_punches\", \"/Users/alanedelman/.julia/packages/Knockout/1sDlc/src/../assets/knockout_punches.js\"), Asset(\"js\", nothing, \"/Users/alanedelman/.julia/packages/InteractBase/9mFwe/src/../assets/all.js\"), Asset(\"css\", nothing, \"/Users/alanedelman/.julia/packages/InteractBase/9mFwe/src/../assets/style.css\"), Asset(\"css\", nothing, \"/Users/alanedelman/.julia/packages/Interact/SbgIk/src/../assets/bulma_confined.min.css\")], Dict{Any,Any}(\"changes\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"index\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"index\\\"]()) ? (this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\")]), WebIO.ConnectionPool(Channel{Any}(sz_max:32,sz_curr:0), Set(AbstractConnection[]), Base.GenericCondition{Base.AlwaysLockedST}(Base.InvasiveLinkedList{Task}(Task (runnable) @0x000000010ae28250, Task (runnable) @0x000000010ae28250), Base.AlwaysLockedST(1))), WebIO.JSString[WebIO.JSString(\"function () {\\n    var handler = (function (ko, koPunches) {\\n    ko.punches.enableAll();\\n    ko.bindingHandlers.numericValue = {\\n        init: function(element, valueAccessor, allBindings, data, context) {\\n            var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n            stringified.subscribe(function(value) {\\n                var val = parseFloat(value);\\n                if (!isNaN(val)) {\\n                    valueAccessor()(val);\\n                }\\n            });\\n            valueAccessor().subscribe(function(value) {\\n                var str = JSON.stringify(value);\\n                if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n                     return;\\n                 if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n                     return;\\n                stringified(str);\\n            });\\n            ko.applyBindingsToNode(\\n                element,\\n                {\\n                    value: stringified,\\n                    valueUpdate: allBindings.get('valueUpdate'),\\n                },\\n                context,\\n            );\\n        }\\n    };\\n    var json_data = {\\\"formatted_vals\\\":[\\\"1\\\",\\\"2\\\",\\\"3\\\",\\\"4\\\",\\\"5\\\",\\\"6\\\",\\\"7\\\",\\\"8\\\",\\\"9\\\",\\\"10\\\",\\\"11\\\"],\\\"changes\\\":WebIO.getval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"}),\\\"index\\\":WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"})};\\n    var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        [this[\\\"formatted_val\\\"]=ko.computed(    function(){\\n        return this.formatted_vals()[parseInt(this.index())-(1)];\\n    }\\n,this)]\\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"17084213940907812968\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n);\\n    (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/2618192f019dfcc40a9013ef31cb09a511d8ea26-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/df29ad18e8884e98d1d36dd0c339f642f729bf96-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol,Any}(:className => \"field interact-widget\")), Observable{Any} with 0 listeners. Value:\n",
       "Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Plot{Plots.GRBackend() n=25}], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\"))], Dict{Symbol,Any}())"
      ]
     },
     "execution_count": 24,
     "metadata": {
      "application/vnd.webio.node+json": {
       "kernelId": "491971e1-5e20-4d24-b021-08a27ca47da3"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for i=slider(1:length(Akeep),value=1)\n",
    "      lookat(Akeep[i],redrow=row[i],showtext=true)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Array{Float64,2}:\n",
       "  1.0       0.0        0.0      0.0      0.0\n",
       " 11.2268    1.0        0.0      0.0      0.0\n",
       "  1.65127   0.100262   1.0      0.0      0.0\n",
       " 22.1736    2.04623   -7.9508   1.0      0.0\n",
       " -7.45666  -0.817478  -7.32798  1.03327  1.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U =  Akeep[end]\n",
    "A =  Akeep[1]\n",
    "L += I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Array{Float64,2}:\n",
       "  0.0737129  -0.934928   0.686715   0.756193   0.665156\n",
       "  0.827557   -0.250188   0.22969   -0.451796  -0.067432\n",
       "  0.12172    -0.516531   0.479388  -1.64916    0.735843\n",
       "  1.63448     0.234953  -0.837002   1.12689   -2.08616 \n",
       " -0.549652   -1.40446    0.295059   2.6386    -1.12414 "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L*U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Array{Float64,2}:\n",
       "  0.0737129  -0.934928   0.686715   0.756193   0.665156\n",
       "  0.827557   -0.250188   0.22969   -0.451796  -0.067432\n",
       "  0.12172    -0.516531   0.479388  -1.64916    0.735843\n",
       "  1.63448     0.234953  -0.837002   1.12689   -2.08616 \n",
       " -0.549652   -1.40446    0.295059   2.6386    -1.12414 "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L*U ≈ A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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